Properties

Label 4032.2.i.b
Level 4032
Weight 2
Character orbit 4032.i
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM No
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.i (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} -\beta_{3} q^{11} -2 \beta_{2} q^{17} + 2 \beta_{7} q^{19} -\beta_{6} q^{23} + q^{25} -\beta_{7} q^{29} + 4 \beta_{4} q^{31} + ( -2 \beta_{3} + \beta_{5} ) q^{35} + \beta_{5} q^{37} -2 \beta_{2} q^{41} + 2 \beta_{1} q^{43} + 8 q^{47} + ( 3 - 2 \beta_{6} ) q^{49} -3 \beta_{7} q^{53} + 2 \beta_{4} q^{55} -2 \beta_{5} q^{59} + 6 \beta_{3} q^{61} -\beta_{1} q^{67} -3 \beta_{6} q^{71} + 2 \beta_{6} q^{73} + ( \beta_{1} - \beta_{7} ) q^{77} -4 \beta_{2} q^{79} + 2 \beta_{5} q^{85} + 6 \beta_{2} q^{89} -4 \beta_{6} q^{95} -2 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} + 64q^{47} + 24q^{49} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 16 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\( 2 \nu^{6} + 12 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} + \nu^{5} + 29 \nu^{3} + 11 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{7} + \nu^{5} - 29 \nu^{3} + 11 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 3 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} - 11 \beta_{4} - 11 \beta_{3}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{5} - 9 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} - 13 \beta_{6} - 29 \beta_{4} + 29 \beta_{3}\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
−0.437016 0.437016i
0.437016 + 0.437016i
1.14412 + 1.14412i
−1.14412 1.14412i
0.437016 0.437016i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
1.14412 1.14412i
0 0 0 2.00000i 0 −2.23607 1.41421i 0 0 0
1889.2 0 0 0 2.00000i 0 −2.23607 + 1.41421i 0 0 0
1889.3 0 0 0 2.00000i 0 2.23607 1.41421i 0 0 0
1889.4 0 0 0 2.00000i 0 2.23607 + 1.41421i 0 0 0
1889.5 0 0 0 2.00000i 0 −2.23607 1.41421i 0 0 0
1889.6 0 0 0 2.00000i 0 −2.23607 + 1.41421i 0 0 0
1889.7 0 0 0 2.00000i 0 2.23607 1.41421i 0 0 0
1889.8 0 0 0 2.00000i 0 2.23607 + 1.41421i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.b Even 1 yes
12.b Even 1 yes
21.c Even 1 yes
24.f Even 1 yes
28.d Even 1 yes
56.e Even 1 yes
168.i Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{47} - 8 \)